Your last photo appears to be of the end of the camshaft under the valve cover - with the distributor removed. Did you remove the distributor? If so, why? For the items you replaced, you wouldn't remove the distributor.

For the distributor cap and rotor (assuming the distributor is original to that engine and car), what exactly did you buy (what manufacturer and what part number)?

 

Your last photo appears to be of the end of the camshaft under the valve cover - with the distributor removed. Did you remove the distributor? If so, why? For the items you replaced, you wouldn't remove the distributor.

For the distributor cap and rotor (assuming the distributor is original to that engine and car), what exactly did you buy (what manufacturer and what part number)?

I ended up taking the distributor out after the car didn’t turn on when I replaced the plug wires and the rotor.
My step dad insisted that I had timed it wrong and that’s the reason why the car won’t start, but I installed everything correctly and showed him that I had done everything correctly and that it might be something else causing the issue.

The plug wires I got were a duralast gold set and it included a new distributor cap, the part number is A801253 and the ignition rotor's part number is DR925

 

I would assume your new parts aren't compatible. I recently changed the distributor on my 1996 celica and I made sure to keep the rotor pointing the same angle. Car started up fine. You want to make sure that the distributor is properly grounded and that it makes proper contact with the engine block.

 

I have some free time today so I’ll do that and get back to you

 

Maybe a dumb question, but do you have the spark plug wires on the correct distributor cap terminals?

I added the photo below. The distributor is for a 1992-1993 5S-FE, but the firing order is the same as your 3S-FE (1-3-4-2), and I believe the spark plug wire terminal locations are also the same.

 

It’s not a dumb question you’re good, and yeah I made sure I had the plug wires on the correct cap terminals. (and yes I know that the two distributor terminals are disconnected in this picture, I connected them after the picture was taken)

 

Is it possible that the resistance of your new spark plug wires doesn't match OEM? Have you checked?
You can put the old wires back on and see if that makes any difference. My OEM wires have resistance of about 6 kOhm. The aftermarket ones that I've tried have much lower resistance (though not sure if it matters).

 

Sadly I can’t reuse the Oem plug wires since the terminal that plugs into the spark plug broke on 3 of them when I was replacing them.

 

If you are getting spark and your confident your timing is close, try putting some starting fluid or some gasoline in your intake manifold (through your throttle body) and see if the engine try’s to start it does start and run momentarily. If it does, you have a fuel problem, not an ignition problem.

 

I wonder if you are off 180 deg. The engine is 4 stroke so the piston is at top 2x in its cycle. The other top is after expelling exhaust and before pulling in fuel and air.

Try another revolution. Rotate to top. Pull the distributor, rotate 360 to the next top and reinsert distributor (without adjustment)